Stefan-Boltzmann law - AbsoluteAstronomy.com

The Stefan–Boltzmann law, also known as Stefan's law, states that the total energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

radiated per unit surface area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

of a black body

Black body

A black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum...

per unit time

Time

Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

(also known as the black-body irradiance
Irradiance
Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...

or emissive power), j^*, is directly proportional

Proportionality (mathematics)

In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

to the fourth power of the black body's thermodynamic temperature

Thermodynamic temperature

Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...

T (also called absolute temperature):

The constant of proportionality σ, called the Stefan–Boltzmann constant or Stefan's constant, derives from other known constants of nature. The value of the constant is

where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

. Thus at 100 K the energy flux density is 5.67 W/m², at 1000 K 56,700 W/m², etc.

A more general case is of a grey body, the one that doesn't absorb or emit the full amount of radiative flux. Instead, it radiates a portion of it, characterized by its emissivity

Emissivity

The emissivity of a material is the relative ability of its surface to emit energy by radiation. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature...

The irradiance j^* has dimensions of energy flux (energy per time per area), and the SI

Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units of measure are joule

Joule

The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

s per second per square metre, or equivalently, watt

Watt

The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...

s per square metre. The SI unit for absolute temperature T is the kelvin

Kelvin

The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

. is the emissivity

Emissivity

of the grey body; if it is a perfect blackbody,

. Still in more general (and realistic) case, the emissivity depends on the wavelength,

.

To find the total absolute power

Power (physics)

In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

of energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

radiated for an object we have to take into account the surface area, A(in m²):

The law was deduced by Jožef Stefan

Joseph Stefan

Joseph Stefan was a physicist, mathematician, and poet of Slovene mother tongue and Austrian citizenship.- Life and work :...

(1835–1893) in 1879 on the basis of experimental measurements made by John Tyndall

John Tyndall

John Tyndall FRS was a prominent Irish 19th century physicist. His initial scientific fame arose in the 1850s from his study of diamagnetism. Later he studied thermal radiation, and produced a number of discoveries about processes in the atmosphere...

and was derived from theoretical considerations, using thermodynamics

Thermodynamics

Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, by Ludwig Boltzmann

Ludwig Boltzmann

Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...

(1844–1906) in 1884. Boltzmann considered a certain ideal heat engine

Heat engine

In thermodynamics, a heat engine is a system that performs the conversion of heat or thermal energy to mechanical work. It does this by bringing a working substance from a high temperature state to a lower temperature state. A heat "source" generates thermal energy that brings the working substance...

with light

Light

Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

as a working matter instead of gas. The law is valid only for ideal black objects, the perfect radiators, called black bodies

Black body

. Stefan published this law in the article Über die Beziehung zwischen der Wärmestrahlung und der Temperatur (On the relationship between thermal radiation and temperature) in the Bulletins from the sessions of the Vienna Academy of Sciences.

Integration of intensity derivation

The law can be derived by considering a small flat black body

Black body

surface radiating out into a half-sphere. This derivation uses spherical coordinates, with φ as the zenith angle and θ as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where φ = ^π/₂.

The intensity of the light emitted from the blackbody surface is given by Planck's law :

where

is the amount of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

per unit surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...

per unit time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

per unit solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

emitted in the frequency range between ν and ν + dν by a black body at temperature T
is Planck's constant
is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

, and
is Boltzmann's constant.

The quantity

is the power

Power (physics)

radiated by a surface of area A through a solid angle

Solid angle

The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

dΩ in the frequency range

.

The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body,

To derive the Stefan–Boltzmann law, we must integrate Ω over the half-sphere and integrate ν from 0 to ∞. Furthermore, because black bodies are Lambertian (i.e. they obey Lambert's cosine law

Lambert's cosine law

In optics, Lambert's cosine law says that the radiant intensity observed from a Lambertian surface or a Lambertian radiator is directly proportional to the cosine of the angle θ between the observer's line of sight and the surface normal. A Lambertian surface is also known as an ideal diffusely...

), the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle φ, and in spherical coordinates, dΩ = sin(φ) dφ dθ.

Then we plug in for I:

To do this integral, do a substitution,

which gives:

The integral on the right can be done in a number of ways (one is included in this article's appendix) – its answer is π⁴/15, giving the result that, for a perfect blackbody surface:

An alternative form of the Stefan–Boltzmann constant, more fundamental to physics:

Finally, this proof started out only considering a small flat surface. However, any differentiable surface can be approximated by a bunch of small flat surfaces. So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface; and the total surface area is just the sum of the areas of each surface—so this law holds for all convex

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

blackbodies, too, so long as the surface has the same temperature throughout.

Thermodynamic derivation

The fact that the energy density of the box containing radiation is proportional to

can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure

is related to the internal energy density:

The total internal energy of the box containing radiation can thus be written as:

Inserting this in the fundamental thermodynamic relation

yields

This equation can be used to derive a Maxwell relation

Maxwell relations

Maxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of...

. From the above equation it can be seen that:

and

The symmetry of second derivatives

Symmetry of second derivatives

In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a functionfof n variables...

with regard to

and

then implies:

Because the pressure is proportional to the internal energy density it depends only on the temperature and not on the volume. In the derivative on the right hand side, the temperature is thus a constant. Evaluating the derivatives gives the differential equation:

This can be solved by integrating with respect to T to give

This implies that

Temperature of the Sun

With his law Stefan also determined the temperature of the Sun

Sun

The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

's surface. He learned from the data of Charles Soret

Charles Soret

Charles Soret was a Swiss physicist and chemist. He is universally known for his work on thermodiffusion .-Life:...

(1854–1904) that the energy flux density from the Sun is 29 times greater than the energy flux density of a warmed metal lamella. A round lamella was placed at such a distance from the measuring device that it would be seen at the same angle as the Sun. Soret estimated the temperature of the lamella to be approximately 1900 °C

Celsius

Celsius is a scale and unit of measurement for temperature. It is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...

to 2000 °C. Stefan surmised that ⅓ of the energy flux from the Sun is absorbed by the Earth's atmosphere

Earth's atmosphere

The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

, so he took for the correct Sun's energy flux a value 3/2 times greater, namely 29 × 3/2 = 43.5.

Precise measurements of atmospheric absorption

Absorption (electromagnetic radiation)

In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...

were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.57⁴ = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of a lamella, so Stefan got a value of 5430 °C or 5700 K (the modern value is 5778 K). This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as 13,000,000 °C were claimed. The lower value of 1800 °C was determined by Claude Servais Mathias Pouillet

Claude Servais Mathias Pouillet

Claude Servais Mathias Pouillet was a French physicist and a professor of physics at the Sorbonne and member of the French Academy of Science.Pouillet was born in Cusance, Doubs, France....

(1790–1868) in 1838 using the Dulong-Petit law. Pouilet also took just half the value of the Sun's correct energy flux.

Temperature of stars

The temperature of star

Star

A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s other than the Sun can be approximated using a similar means by treating the emitted energy as a black body

Black body

radiation. So:

where L is the luminosity

Luminosity

Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...

, σ is the Stefan–Boltzmann constant, R is the stellar radius and T is the effective temperature

Effective temperature

The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation...

. This same formula can be used to compute the approximate radius of a main sequence star relative to the sun:

where

, is the solar radius, and so forth.

With the Stefan–Boltzmann law, astronomer

Astronomer

An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

s can easily infer the radii of stars. The law is also met in the thermodynamics

Black hole thermodynamics

In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons...

of black hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

s in so-called Hawking radiation

Hawking radiation

Hawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...

Temperature of the Earth

Similarly we can calculate the effective temperature

Effective temperature

The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation...

of the Earth T_E by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation. The amount of energy, E_S, emitted by the Sun is given by:

At Earth, this energy is passing through a sphere with a radius of a₀, the distance between the Earth and the Sun, and the energy passing through each square metre of the sphere is given by

The Earth has a radius of r_E, and therefore has a cross-section of

. The amount of solar energy absorbed by the Earth is thus given by:

The amount of energy emitted must equal the amount of energy absorbed, and so:

T_E can then be found:

where T_S is the temperature of the Sun, r_S the radius of the Sun, and a₀ is the distance between the Earth and the Sun. This gives an effective temperature of 6°C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere.

The Earth has an albedo

Albedo

Albedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...

of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.7, but that the planet still radiates as a black body (the latter by definition of effective temperature

Effective temperature

The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation...

, which is what we are calculating). This approximation reduces the temperature by a factor of 0.7^1/4, giving 255 K (−18 °C).

However, long-wave radiation from the surface of the earth is partially reflected (or absorbed and re-radiated back down) in the atmosphere, instead of being radiated away, by greenhouse gases, namely water vapor

Water vapor

Water vapor or water vapour , also aqueous vapor, is the gas phase of water. It is one state of water within the hydrosphere. Water vapor can be produced from the evaporation or boiling of liquid water or from the sublimation of ice. Under typical atmospheric conditions, water vapor is continuously...

, carbon dioxide

Carbon dioxide

Carbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...

and methane

Methane

Methane is a chemical compound with the chemical formula . It is the simplest alkane, the principal component of natural gas, and probably the most abundant organic compound on earth. The relative abundance of methane makes it an attractive fuel...

. Since the emissivity with greenhouse effect (weighted more in the longer wavelengths where the Earth radiates) is reduced more than the absorptivity (weighted more in the shorter wavelengths of the Sun's radiation) is reduced, the equilibrium temperature is higher than the simple black-body calculation estimates. As a result, the Earth's actual average surface temperature is about 288 K (14 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.

Appendix

In one of the above derivations, the following integral appeared:

where

is the polylogarithm function and

is the Riemann zeta function. If the polylogarithm function and the Riemann zeta function are not available for calculation, there are a number of ways to do this integration; a simple one is given in the appendix of the Planck's law article. This appendix does the integral by contour integration. Consider the function:

Using the Taylor expansion of the sine function, it should be evident that the coefficient of the k³ term would be exactly -J/6.
By expanding both sides in powers of

, we see that

is minus 6 times the coefficient of

of the series expansion of

. So, if we can find a closed form for f(k), its Taylor expansion will give J.

In turn, sin(x) is the imaginary part of e^ix, so we can restate this as: